Aggregation and Breakup of Colloidal Particle ... - ACS Publications

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Langmuir 2004, 20, 2614-2627

Aggregation and Breakup of Colloidal Particle Aggregates in Shear Flow, Studied with Video Microscopy V. A. Tolpekin, M. H. G. Duits,* D. van den Ende, and J. Mellema Faculty of Science and Technology, Physics of Complex Fluids Group, Associated with the J.M. Burgerscentrum for Fluid Mechanics, and Institute of Mechanics, Processes and ControlsTwente (IMPACT), University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands Received September 19, 2003. In Final Form: January 26, 2004 We used video microscopy to study the behavior of aggregating suspensions in shear flow. Suspensions consisted of 920 nm diameter silica spheres, dispersed in a methanol/bromoform solvent, to which poly(ethylene glycol) (M ) 35.000 g) was added to effect weak particle aggregation. With our solvent mixture, the refractive index of the particles could be closely matched, to allow microscopic observations up to 80 µm deep into the suspension. Also the mass density is nearly equal to that of the particles, thus allowing long observation times without problems due to aggregate sedimentation. Particles were visualized via fluorescent molecules incorporated into their cores. Using a fast confocal scanning laser microscope made it possible to characterize the (flowing) aggregates via their contour-area distributions as observed in the focal plane. The aggregation process was monitored from the initial state (just after adding the polymer), until a steady state was reached. The particle volume fraction was chosen at 0.001, to obtain a characteristic aggregation time of a few hundred seconds. On varyiation of polymer concentration, cP (2.2-12.0 g/L), and shear rate, γ (3-6/s), it was observed that the volume-averaged size, Dv, in the steady state became larger with polymer concentration and smaller with shear rate. This demonstrates that the aggregate size is set by a competition between cohesive forces caused by the polymer and rupture forces caused by the flow. Also aggregate size distributions were be measured (semiquantitatively). Together with a description for the internal aggregate structure they allowed a modeling of the complete aggregation curve, from t ) 0 up to the steady state. A satisfactory description could be obtained by describing the aggregates as fractal objects, with Df ) 2.0, as measured from CSLM images after stopping the flow. In this modeling, the fitted characteristic breakup time was found to increase with cP.

I. Introduction Understanding the flow behavior of weakly aggregating particle dispersions is important for many different applications (e.g., food and sanitary products, coatings), involving both the formulation and the final application of the fluid. This importance is also evidenced by a body of research literature. Herein, different levels of approach are found. On one end of the spectrum there are empirical descriptions of the flow curve (e.g. shear thinning, yield behavior) as a function of shear-stress or -rate and fluid composition (e.g., ref 1). On the other end, there are modeling studies aimed at describing and predicting the flow behavior on the basis of assumed particle interactions and aggregate structures.2-5 While the former approach usually results in a “working knowledge” only for specific fluids, the latter approach gives a more generic insight but with limited applicability. At present there is still a large gap to bridge between these two approaches. Both the types of aggregating systems and the different mechanisms for structure (de-) formation may have to be sorted out first. The state of the art is that little is known (and hence many challenging questions remain) regarding the mechanisms for structure (de)formation. Weakly aggregating dispersions are “living * To whom correspondence may be addressed. (1) Macosko C. W. RHEOLOGY, principles, measurements, and applications; VCH Publishers: New York, 1994. (2) De Rooij, R.; Potanin, A. A.; Van den Ende, D.; Mellema, J. J. Chem. Phys. 1993, 99, 9213-9223. (3) Potanin, A. A.; Derooij, R.; Van den Ende, D.; Mellema, J. J. Chem. Phys. 1995, 102, 5845-5853. (4) Wolthers, W.; Duits, M. H. G.; vandenEnde, D.; Mellema, J. J. Rheol. 1996, 40, 799-811. (5) Barthelmes, G.; Pratsinis, S. E.; Buggisch, H. Chem. Eng. Sci. 2003, 58, 2893-2902.

systems” in the sense that they keep on evolving, toward an equilibrium state (which usually is not reached). In flow, different possibilities exist for the formation, breakup or restructuring of aggregates. Aggregation may be enabled by Brownian and/or convective transport of the involved entities toward each other. Breakup may be thermally and/or flow induced, and different fragmentations of the same aggregate may be possible. Rearrangements of particles within an aggregate may occur through breakup events, but possibly also via sliding or rolling (i.e., without loss of interparticle contact). Previous Research. Studies of aggregation (and breakup) in shear flow, using microscopy and scattering techniques to resolve the structure of aggregating particle systems, have been performed over the years and are still being applied up to date (see refs 6-11 and references therein). The following section highlights some microscopy studies that are most closely related to the present paper. Van de Ven and Mason in one of their seminal papers in 1977 9 (part VII in the series on “the microrheology of colloidal dispersions”) describe experiments aimed at the initial stages of aggregation of 1 µm PS latex spheres in a precision-bore glass microtube. They compared the distribution of the particles over singlets, doublets, and (6) Oles, V. J. Colloid Interface Sci. 1992, 154, 351-358. (7) Torres, F. E.; Russel, W. B.; Schowalter, W. R J. Colloid Interface Sci. 1991, 142, 554-574. (8) Selomulya, C.; Amal, R.; Bushell, G.; Waite, T. D J. Colloid Interface Sci. 2001 236, 67-77. (9) van de Ven, T. G. M.; Mason, S. G. Colloid Polym. Sci. 1977, 255, 468-479. (10) Hoekstra, H.; Vermant, J.; Mewis, J. Langmuir 2003, 19, 91349141. (11) Hamberg, L.; Walkenstrom, P.; Stading, M.; Hermansson, A. M. Food Hydrocolloids 2001, 15, 139-151.

10.1021/la035758l CCC: $27.50 © 2004 American Chemical Society Published on Web 02/27/2004

Aggregating Suspensions in Shear Flow

triplets (and higher) at the entrance and the exit of the tube, for different shear rates in the high Peclet number regime. In the 1990s, video microscopy techniques started to become increasingly applied to study structure formation in colloidal fluids. An overview can be found in ref 12. De Hoog et al.13 used CSLM to study the kinetics and the morphology of the phase separation (in the absence of flow) in a mixture of colloidal PMMA particles and nonadsorbing polymer. Digital image processing was used to determine the dependence of the characteristic aggregation rate and the size of the clusters on the polymer concentration. Hoekstra et al.10 used video microscopy to study the time evolution of flow-induced changes in the structure of two-dimensional suspensions. Their system consisted of 2.5 µm sized PS latex particles, suspended at an airaqueous liquid interface. Shear and extensional flows were imposed on strongly and weakly aggregated systems, and the recordings were analyzed for averaged coordination number and anisotropy. One remark of the authors is that “Confocal microscopy experiments on model colloidal systems are promising, but time-resolved measurements remain difficult.” Hamberg et al.11 used a CSLM in combination with a four-roll mill to study the timedependent aggregation of WPI-coated 5 micron PS latex spheres in a (steady) hyperbolic flow-field. A correlation between the largest aggregate size and macroscopic viscosity data was made, albeit with limited time resolution as these authors remark. Varadan and Solomon14 used CSLM to study gels of 832 nm diameter hydrophobized silica particles. The static structures of the gels were characterized via the radial distribution function and via the probability distribution of Voronoi polyhedra volumes. One remark of these authors is that “the density mismatch of silica particles with typical solvents, limits their use for characterization of fluid-state structure”. In a followup study, Varadan and Solomon15 studied the influence of squeeze flow on the formation of voids and cracks. Microscopic studies on aggregation in flow are not limited to synthetic colloidal particle systems. Goldsmith et al.16 used a Couette shear cell in combination with highspeed video microscopy to study the shear rate dependence of the aggregation of neutrophils (8 µm). A remarkable dependence of aggregation and breakup on shear rate (shear stress) was found, with a complete aggregate breakup below a critical shearrate. Also, modes of fragmentation like the loss of single or multiple neutrophils could be characterized. The Present Work. In the current paper we present observations on strongly fluorescent, aggregating colloidal spheres in shear flow, using a fast confocal scanning laser microscope. With this combination of fluid and instrument, new possibilities emerged: the observation of large numbers of aggregates in real space and time. This allowed us to measure aggregate size distributions as a function of time and to observe structure parameters of the aggregates. The experiments were performed with 920 nm diameter silica spheres, which were made to form weak aggregates after adding a nonadsorbing polymer. To optimize possibilities for observation, we composed a solvent mixture which allows matching the silica particles, (12) Habdas, P.; Weeks, E. R. Curr. Opin. Colloid Interface Sci. 2002, 7, 196-203. (13) de Hoog, E. H. A.; Kegel, W.K.; van Blaaderen, A.; Lekkerkerker, H. N. W. Phys. Rev. E 2001, 64, Art. No. 021407. (14) Varadan P.; Solomon, M. J. Langmuir 2003, 19, 509-512. (15) Varadan, P.; Solomon, M. J. J. Rheol. 2003, 47, 943-968. (16) Goldsmith, H. L.; Quinn, T. A.; Drury, G.; Spanos, C.; McIntosh, F. A.; Simon, S. I Biophys. J. 2001, 81, 2020-2034.

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regarding their refractive index as well as their mass density. The (near) zero buoyancy allowed long enough experiments to reach a “steady-state” aggregate size distribution. One particular focus was the breakup behavior of aggregates in shear flow. Although model concepts have been presented, and in cases have given a quantitative description of rheological experiments (e.g., refs 2 and4), validation is lacking. One breakup criterion which has been successfully applied to describe the flow curve of weakly (polymer induced) aggregating particles in shear flow has been a mechanical equilibrium between cohesive and disruptive forces.4 In this picture, a critical aggregate is at the brink of being broken into two equal parts. These fragments experience a cohesive force due an unbalanced osmotic pressure of the polymer and a separation force caused by the differently directed hydrodynamic drags. Since this (model) breakup force grows more rapidly with aggregate size than the (model) cohesive force, an upper limiting aggregate size was predicted. Considering the origins of the forces, the maximum aggregate size should increase with polymer concentration and decrease with shear rate. Choosing polymer concentration and shear rate as experimental variables, as we did, allows verification of this concept. More precisely, breakup should not be defined by a mechanical equilibrium but by kinetic coefficients, which have a certain dependence on aggregate size and shear rate. Detailed information obtained in this study describes breakup at the level of kinetic coefficients. This was achieved by setting up a population balance model, which was fitted to the entire aggregation curve (i.e., from t ) 0 up to the steady state). Input parameters in this model were the experimentally obtained aggregate size (distribution)s as a function of time and the fractal dimension as measured for typical aggregates. This paper is further organized as follows. In section II, the kinetics of aggregation and breakup will be described with a population balance model. In section III, the development of the aggregating fluid and the experiments in the CSLM shear cell will be discussed. Since this involves a new experimental method, we have chosen to give a detailed account of issues encountered. In section IV the obtained aggregation curves, internal aggregate structures, and size distributions will be presented and analyzed, while in section V the conclusions will be drawn. II. Theory: The Rate Equation for the Average Aggregate Size In this section, we shall describe the kinetics of aggregation and breakup, to arrive at a description of the (time-dependent) aggregation curve. Assuming the aggregates to be fractal like, the number N of primary particles inside an aggregate scales with its radius RN as

N ) No(RN/a)Df a RN ) a(N/No)1/Df

(II.1)

where Df is the fractal dimension, a the radius of a particle, and No is a constant of order unity. The number density of aggregates containing N particles is written as n(N) and the total number density of aggregates as Nagg. Thus the volume fraction of particles is given by: φp ) (4/3)πa3np ) (4/3)πa3∑Nn(N) ) (4/3)πa3〈N〉Nagg, where 〈N〉 ) ∑Nn(N)/∑n(N) is a number average and Nagg ) ∑n(N); np is the number density of particles. The time evolution of the number density n(N) is described by the population balance equation:

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∂

Langmuir, Vol. 20, No. 7, 2004

1

Tolpekin et al.

N-1

∑ B(N-M,M) n(N-M) n(M) 2 M)1

N˙ agg ) -

∑ B(N,M) n(N) n(M) - K(N) n(N) + M)1

M

n(N,t) )

∂t ∞

(II.2)

with B(N,M) the kernel describing the aggregation flux between aggregates containing N and M particles, respectively: B(N,M) ) (4/3)γ˘ qNM3, with γ˘ the shear rate and qNM the center to center distance between the colliding aggregates when they touch: qNM ) κ(RN + RM) with κ a dimensionless constant slightly less than unity, representing the “hydrodynamic softness” of an aggregate. The choice to let B depend on (N,M) reflects the assumption that aggregates will grow via binary collisions only. For shear controlled transport B(N,M) becomes

B(N,M) )

( ) (( ) 4 N γ˘ a3κ3 3 No

1/Df

+

( ) ) M No

1/Df 3

(II.3)

which illustrates how B(N,M) depends explicitly on the fractal dimension after the substitution of qNM. The kernel K(N) is the rate of breakup for an aggregate containing N particles. It is modeled as the breakup rate for a two-particle bond, 1/tesc, times the number of twoparticle bonds inside an aggregate. As an ansatz, This number is assumed to scale with (R/a)Q and so

(RN/a)Q ) Ko(RN/a)Q K(N) ) tesc

The function p(N;M) is the probability that an aggregate containing M particles, when it breaks to pieces, will break into a fragment containing N and one containing M - N particles. The function p(N;M) must satisfy the following M-1 p(N;M) ) 2 because we assume that conditions: (a) ∑N)1 each parent aggregate breaks into two daughters, (b) M-1 Np(N;M) p(N;M) ) p(M-N;M) by definition, and (c) ∑N)1 ) M because the total number of particles in the system must be conserved. In principle the time evolution of the distribution n(N,t) can be calculated by solving eq II.2 numerically and from this the time dependence of quantities such as the total number of aggregates or the average aggregate size. However, if the distribution n(N,t) is self-preserving, these quantities can be obtained more directly. The time rate of change of the total number of aggregates is obtained by integration of eq II.2 ∞

N˙ agg )

∑

N)1

∂n(N) ∂t

( )∑ ∑

)-

1

2

∞

-

(( )

∞

N

1/Df

+

No

()) ( ) (〈( ) 〉 〈( ) 〉〈( ) 〉) 1/Df 3

4

3

∞

n(N) n(M) + Ko N

γ˘ a3κ3Nagg2

3/Df

No

∑ f(N) n(N) ) N)1

+3

N

1/Df

No

N

2/Df

+ No KoNagg〈f(N)〉

where f(N) ) (RN/a)Q. Moreover since φp ) (4/3)πa3〈N〉Nagg, one can express Nagg in 〈N〉

Nagg )

3φp

1 4πa3 〈N〉

and

N˙ agg ) -

3φp 1 d Nagg d 〈N〉 ) 〈N〉 dt 〈N〉 4πa3 〈N〉2 dt

from which the rate equation for the average aggregate size follows as

(〈( ) 〉 〈( ) 〉〈( ) 〉)

γ˘ κ3φp d 〈N〉 ) dt π

N No

3/Df

+3

N No

1/Df

N No

2/Df

-

Ko〈N〉〈f(N)〉

With eq II.1, 〈N〉 ) N〈(R/a)Df〉, one obtains

γ˘ φpκ3 d 〈(R/a)Df〉 ) (〈(R/a)3〉 + 3〈R/a〉〈(R/a)2〉) dt πNo Ko〈(R/a)Df〉〈(R/a)Q〉 (II.4) For monodisperse aggregates, 〈(R/a)S〉 ) (R/a)S, eq II.4 reduces to

4γ˘ φpκ3 d Df (R/a) ) (R/a)3 - Ko(R/a)Df+Q dt πNo

(II.5)

which leads, with Bo ) (4γ˘ φpκ3)/(πNo) to

1 d (R/a) ) (Bo(R/a)4-Df - Ko(R/a)Q+1) dt Df

(II.6)

The solution of eq II.6 can be used to describe (i.e., fit) experimentally observed aggregation curves by choosing trial values for Bo (in a small range), tesc, and Q. R(t) and Df can be determined from the recordings. For our system (φp ) 0.001, κ = 0.9, No = 0.75) we can estimate: Bo = (1.25 × 10-3)γ˘ . In the steady state the left-hand side of eq II.6 becomes zero, and size of the aggregates is given by

Bo(R/a)4-Df ) Ko(R/a)Q+1 or

∞

B(N,M) n(N) n(M) +

N)1 M)1

Ko/Bo ) (R/a)3-df-Q

∞

∑ K(N) n(N) N)1 Thus

∞

∑ ∑ γ˘ a3κ3 3 N)1 M)1

No

∞

∑ K(M) n(M) p(M;N) M)N+1

2

In fact eq II.6 is more general than stated. If we suppose self-preservation of the number distribution n(N,t), i.e., n(N,t) ) fnc(N/Nsc(t)), the averages 〈(R/a)S〉 will scale with 〈R/a〉S where the scaling factor will depend on the function fnc(x) but not on the time, as can be concluded from the


Langmuir, Vol. 20, No. 7, 2004 2617

Figure 1. Mass density and refractive index of bromoform/methanol mixtures as a function of bromoform volume fraction (x) at 25 °C. Dashed lines result from data fitting with a linear function. Mass density: F ) 2.023x + 0.876 (g/mL). Refractive index: n ) 0.261x + 1.328.

following observation

〈A(t)〉 ) 〈B(t)〉

∫0∞ AN n(N,t) dN ) ∫0∞ BN n(N,t) dN ∞ Nsc(t) ∫0 AN fnc(xN) dxN 〈A(0)〉 ) ∞ Nsc(t) ∫0 BN fnc(xN) dxN 〈B(0)〉

with xN ) N/Nsc(t) and dN ) dxN Nsc(t). Therefore we rewrite eq II.6 as

1 d 〈R/a〉 ) (β1Bo〈R/a〉4-Df - β2Ko〈R/a〉Q+1) dt Df

(II.7)

where β1 and β2 are defined as

β1 )

〈(R/a)3〉 + 3〈R/a〉〈(R/a)2〉 4〈R/a〉3-D〈(R/a)D〉

and

β2 )

〈(R/a)Q〉 〈R/a〉Q

The solution of eq II.7 is equal to that of eq II.6 if one replaces Bo by β1Bo and Ko by β2Ko. Equation II.7 can be integrated numerically, resulting in the time evolution of the average size distribution 〈R(t)/a〉. III. Experiments III.1. Synthesis of Silica Particles. The method of Sto¨ber et al.17 was used to synthesize (primary) silica particles from tetraethyl ortho silicate (TEOS) in the presence of ammonia, water, and ethanol. To make the particles visible with the fluorescence confocal microscope, also a relatively small amount of dye was added to the TEOS in the first preparation step. The dye was fluorescein isothiocyanate (FITC), coupled to (aminopropyl)triethoxysilane (APTES) as described by Verhaegh et al.18 The radius of the thus obtained fluorescent spheres amounted 250 nm. These primary particles were grown out to larger spheres via seeded growth, thereby mimicking the continuous addition method as described by Giesche,19 which (17) Sto¨ber, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968, 26, 62. (18) Verhaegh, N. A. M.; van Blaaderen, A. Langmuir 1994, 10, 14271438. (19) Giesche, H. J. Eur. Ceram. Soc. 1994, 14, 205-214.

minimizes secondary nucleation. More details can be found in Tolpekin et al.20 Transmission electron microscopy (TEM) results showed a particle radius of 460 nm and a polydispersity (standard deviation) of 8%. III.2. Solvent Mixture. To maximize the possibilities for observing the formation and restructuring of the particle aggregates (distribution) with CSLM, a new dispersion medium was sought. Besides the ability to keep the silica particles dispersed as individual entities, the new medium had to meet the requirements that both the refractive index and the mass density should be closely matched to that of the particles. The former is needed to minimize light scattering effects, which can seriously obscure particle observation with the CSLM (reduction of the “working distance” range). The mass-density matching is needed to slow particle/aggregate sedimentation (and hence increase the “working time” for measurements). The refractive index n of the silica particles was assumed to be 1.45, based on previous (lightscattering-contrast variation) experiments on similar systems. The particle mass density F was determined by measuring the particle mass fraction and the overall mass density of a dispersion and was found to be 1.90 g/mL. By measuring mass densities and refractive indices for selected solvent mixtures, we found that solvents containing approximately equal volumes of bromoform (Br) and methanol (Me) should meet the requirements very well; see Figure 1. This prediction, based on interpolating the found n(x) and F(x) dependences with straight lines, was confirmed by inspecting particle dispersions at various solvent compositions for their transparency (with the naked eye) and for their sedimentation or creaming velocity (centrifuging at 100g to increase the buoyancy forces). For the later performed CSLM experiments the solvent composition was chosen such that the particles had an excess density of 0.01 g/mL (81.3 mass % of bromoform). This was done to ensure that any large aggregates formed in the fluid will show up at the bottom of the cell, where the CSLM observations are made. The margin of 0.01 g/mL was chosen to cover for temperature variations (up to 5 °C) as well as for small changes in solvent composition over time (e.g., due to evaporation). III.3. Colloidal Stability in the Solvent. The 460 nm (radius) silica particles turned out to be colloidally stable (i.e., dispersed as single entities) for prolonged times in bromoform/methanol mixtures. The sedimentation velocity of the solvent-dispersion interface was found to be in agreement with the particles being present as single entities (see Tolpekin et al.20 for a more detailed discus(20) Tolpekin, VA.; Duits, M. H. G.; van den Ende, D.; Mellema, J. Langmuir 2003, 19, 4127-4137.

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sion). Sediments that were formed after a week could be easily redispersed by shaking the tube (creamed material could not be fully redispersed, as a consequence of air exposure). We remark here that the precise details of the particle interaction potential are yet unknown for silica particles in bromoform/methanol solvent. For smaller silica particles (25 nm radius) at high concentration, we observed gelation in Br/Me solvent mixtures.21 III. 4. Polymer-Induced Flocculation. Particle aggregation with a controllable strength was effected by adding polymer to the dispersions in Br/Me. Poly(ethylene glycol) (PEG) with a molar mass of 35.000 g/mol was chosen for this, based on its solubility in Br/Me. Before the PEG was added, a small amount of DMF (1%) was supplemented to the dispersion. DMF is known to strongly adsorb at silica surfaces and thus act as a “displacer”, i.e., to counteract polymer adsorption.22 Variation of the polymer concentration between 2 and 12 g/L (at a particle volume fraction of 0.001) showed the occurrence of a “rapid” aggregation and subsequent sedimentation (much faster than the time needed for single particles to settle) for PEG concentrations above a “critical concentration”, which was between 1.8 and 2.2 g/L. We note that this “critical concentration” is based on observations with the microscope and defined by the highest PEG concentration where there is still a complete absence of aggregates. It is not a critical concentration in the macroscopic thermodynamic sense. Diluting aggregated dispersions with Br/Me, so as to lower the PEG concentration well below 2 g/L, resulted in complete disintegration of the aggregates. This demonstrates the absence of “irreversible” (i.e., under no circumstance reversible) aggregation, as would have occurred in the case of a strong adsorption of the polymer at the silica surface (“bridging flocculation” see, e.g., Napper23). Hence it is indicated that the particle attractions are governed by the osmotic pressure of the polymer, as in the case of “polymer depletion”. The strength of the attractions can be estimated by considering the polymers as hard spheres, with a radius as = rg (the radius of gyration) and volume fraction φpol ) (4π/3)npolrg3 (with npol the number density). The attraction between two silica particles at contact is then approximately given by

U)-

3a φ kT 2 rg pol

and

F=

3 a U )φ kT 2rg 4 r 2 pol g

Taking 9 nm as an estimate for the radius of gyration, one obtains at cpol ) 4 g/L, a φpol of 0.2, corresponding to U ) -17 kT and F ) -4 pN. At the “critical polymer concentration” the contact energy would amount roughly 8 kT. Hence according to a hard-sphere model, the attraction between the silica spheres should be weak and short ranged. A more detailed characterization of the pair potential is beyond the scope of this paper. III.5. Sample Preparation. Five components had to be mixed in order to obtain the aggregating fluid: silica particles, DMF, methanol (Me), bromoform (Br), and PEG. (21) Tolpekin, V.A. Unpublished results. (22) Cohen Stuart, M. A. Private communication. (23) Napper, D. H. Polymeric stabilization of colloidal dispersions; Academic Press: London, 1983.

Tolpekin et al.

In developing a protocol for the mixing, several aspects had to be taken into account: (1) Silica particles flocculate on prolonged exposure to high Br concentration, since Br is a nonsolvent for the particles. (2) PEG does not dissolve in pure Br. (3) Once silica particles and PEG polymer are present together in the same fluid, flocculation will occur. (4) Preferably the particles should be in the presence of DMF before adding the PEG, to prevent (initial) polymer adsorption. (5) Concentrated silica particle stocks should not be exposed to evaporation, since dried particles lose their colloidal stability. (6) All fluids containing bromoform should not be exposed to daylight, to avoid Br decomposition (evidenced by a yellow coloration). Samples were eventually prepared by mixing three stock liquids: (I) a 4% silica dispersion in Br/Me/DMF; (II) a 30 g/L PEG solution in Br/Me; (III) Br/Me. Each of these stocks was prepared to have a density of 1.90 g/mL, i.e., equal to the particle density. The required quantities for each of the stocks were calculated from an equal number of setpoints: the silica and PEG concentrations and the sample volume. In these calculations it was assumed that there were no excess mixing volumes. Samples were prepared in 6-mL capped vials. First stock III was added to the vial, then stock I, and thereafter stock II. After each mixing step the vial was shaken gently by hand, to disperse the particles with respect to the polymers. The viscosity of the samples amounted typically 2 mPa‚s. Just before introduction of the sample into the CSLM cell, a calculated small amount of Me was added to lower the density with 0.01 g/mL (see the foregoing). For samples that had been prepared more than a day before the experiment (see below), this amount occasionally included a compensation for slight evaporation (based on measured weight loss and ascribed exclusively to Me). Introduction of the sample into the cell typically lasted 1 min. It is noteworthy that the mechanical mixing as described was never found to have enhanced aggregation as a net result; i.e., observing freshly mixed samples with CSLM always showed predominantly single particles. Moreover, aggregated samples prepared long (more than a day) in advance and reshaken shortly before introduction into the CSLM, were shown to be indistinguishable (with the CSLM) from the freshly mixed samples. Apparently the aggregates are so weak that even a gentle shaking is sufficient to break them down into single particles. III.6. On-Line Observations with CSLM. An UltraView CSLM System (Perkin-Elmer) was used for imaging the (fluorescent) particles. This system comprises (among others) an Nikon Eclipse TE200 inverted microscope, supplied with a 100×, N.A. 1.30 immersion oil objective. Using a Physik Instrumente P-721.17 piezo tube with a PZ 73E high-speed Z axis controller, the vertical displacement of the objective (and with it, the Z-location in the sample) can be set with submicrometer resolution. Excitation of the FITC fluorescence is done with the 488 nm line of a Melles-Griot 643-PEYB-A01 krypton/argon laser. The image from the Yokogawa CSU10 confocal head is recorded with a Hamamatsu IEEE 1394 C4742-9512ERG camera. This digital CCD camera has 1344 × 1024 pixels. Pixel binning ×2 allowed scanning rates up to =15 images per second. The corresponding pixel size amounted to (2 × 0.0645) µm. Images from the camera were directly stored on a PC hard disk.


Langmuir, Vol. 20, No. 7, 2004 2619

Figure 3. Examples of smeared images at γ ) 3.5/s. Images a and b are taken 30 µm above the bottom plane, while it was 50 µm for images c and d. All images are 87 µm × 66 µm.

Figure 2. Schematic drawing of the central part of the experimental setup. Note that the CSLM is here represented by the objective only.

A home-built shear cell with plate-plate geometry was mounted on top of the inverted microscope of the CSLM; see Figure 2. The bottom of the shear cell was made of 0.17 mm thin glass to allow the observations. All other parts were made of stainless steel to prevent corrosion (bromoform reacts with certain metals, for example, aluminum). Steady rotation of the upper plate established a shear flow of the fluid in the cell. Using a Mattke 2842/ T505/PG30 motor, the rotational speed could be gradually varied between 0 and 2 rpm with steps of 0.001 rpm. All experiments were done at the maximal rotational speed of 2 rpm. The gap width was measured with a micrometer and set to 300 µm ((25 µm) for all experiments. The diameter of the sample compartment was 30 mm. The stage with the shear cell could be moved relative to the objective of the microscope in a horizontal direction, thus allowing observations at different radial positions in the shear cell. The actual radial position was varied between 0 and 9 mm (resulting in variation of the shear rate between 0 and 6.3 s-1) and was measured with an accuracy of 0.5 mm. To prevent evaporation of the solvent, a vapor lock filled with n-hexadecane was used. About 1 mL of dispersion was prepared for each experiment. The shear cell was first filled and then closed by lowering the upper plate with the vapor lock mounted on it. A small hole in the vapor lock allowed for pressure equalization. This hole was closed with a PTFE stopper directly hereafter. Next the microscope was focused, and the recordings were started. Recordings were done in the “temporal mode” of the UltraView software. The objective was fixed at a certain radial (5 mm) and Z-position, and images were captured with the smallest possible time interval (“75 ms”). Each image in a video series was a two-dimensional 87 × 66 µm2 CSLM image. A typical Z-position for recording was 30 or 50 µm, which is substantially bigger than the aggregate sizes, and thus far enough from the bottom to ensure that the observations are done in the bulk. A time interval between short video series (typically 400 frames per series) was introduced to choose an optimum between

not loosing information on one hand and not having to store huge amounts of data on the other hand. In a typical experiment approximately 50 GB of pictures were recorded. The time interval between short video series was gradually changed during the experiment. In the beginning of the experiment, where quick changes were expected, it was set to zero. Later it was increased up to 300 s in some cases. Photobleaching was not important because the flow continuously refreshed the set of particles in the observation window. To check the spatial homogeneity of the sample during the experiment, we also recorded images at different Z-positions (0-80 µm) periodically. After observation of the aggregation curve at one radial position, we also made recordings of the aggregate size(s) reached at two other radial positions (3 and 8-9 mm). III.7. Image Analysis. Images were analyzed with Optimas 6.51 software. As a first step, a 3 × 3 trimmed median filter was applied to smoothen the images. III.7.1. Recordings in the Quiescent State. Particlecontaining entities were identified from the intensity map in a sequential manner. After the maximum intensity, I0, was located, a two-dimensional (2D) Gaussian fit was made to the intensity profile and the corresponding fit parameters (I0,σ,r b) were stored on file. Then the fitted profile was subtracted from the real intensity, thus “deleting” the particle from the image. With the new intensity map, the procedure was repeated, resulting in the identification of the second (third, and so on) brightest particle of the original image. This particle identification procedure was continued until the highest remaining maximum intensity had dropped below 10% of the first found value of I0. The remaining intensity profiles on the map were not counted as particles. We refer to this procedure as the “first thresholding”. The (I0,σ,r b) data sets thus created had a substantially smaller size than the images they were obtained from. III.7.2. Recordings in Flow. Most images were recorded in flow, with particle (aggregate) velocities amounting typically to several tens of µm/s. In this situation, considerable “smearing out” occurs, even for the shortest exposure time from the menu (“75 ms”), and it is no longer possible to distinguish particles within aggregates. Typical results are shown in Figure 3.

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While the number of particles within an aggregate obviously cannot be measured anymore from such smeared images, a determination of the overall aggregate size was still feasible. Besides the loss of information, the recording of aggregates moving at high speeds also gave an advantage: a large flux of aggregates through the observation window of the microscope. This large flux made an important contribution to the statistical significance of the measured aggregate size distributions. Conversely, in absense of flow, the number of aggregates that can be observed per unit time is small. Besides that, the aggregates also grow larger since there is no flowinduced breakup anymore. As a result, aggregates formed in the quiescent state sediment rapidly to the bottom of the cell, where they are forced by gravity to assemble into a network. In our case, this precluded a comparison between the aggregate structure formations at rest and in shear flow. III.7.2.1. Data Processing. Two operations were performed: a correction for the motion unsharpness (smearing), and a rejection of off-focus aggregates. Correction for Motion Unsharpness. The degree of smearing, defined as the elongation of the particle/ aggregate in the direction of flow, depends on the velocity of the aggregates v and the exposure time δt. While v and δt have certain expectation values, we have applied a more precise correction by measuring them empirically. The following procedure was applied: In the first step, an intensity threshold was applied (typically 20% of the maximum pixel intensity for singlets). Next, the images were rotated such that the direction of particle (or aggregate) displacement coincided with a major axis (X or Y). The pertaining rotation angle turned out to be independent of the considered aggregate, as was to be expected for affine motion (i.e., all entities moving along laminar flow lines as dictated by the rotating and the nonrotating plate). Hereafter the (smeared) contour area, S′, for each entity was measured in pixel units and converted to real dimensions. Also the maximum extension of the fluorescent area, both along the flow direction (d|′) and perpendicularly (d⊥′) were measured. After that, the nonsmeared dimensions (d|, d⊥, S) of aggregates were calculated using simple geometrical relations for parallel translation

d⊥ ) d⊥′

Tolpekin et al.

Figure 4. Illustration of how the true area of an aggregate S (in the focal plane) is calculated from its apparent area S′ in an image with “motion unsharpness”. The apparent length increase of a single particle Ds can be used to correct all objects in the same velocity plane.

Figure 5. Degree of smearing (as defined by Figure 4) for single particles in flow. The velocity v is obtained by measuring displacements of (smeared) particles between consecutive images and dividing it by the precise time lapse between two images (typically 75 ms). The slope of the plot yields the real exposure time of 58 ms.

d| ) d|′ - Ds

The reconstructed contour area S can be used as a measure for the aggregate size. We chose to convert it to an equivalent diameter

S ) S′ - d⊥DS

D ) (4S/π)1/2

where Ds was determined by measuring d|′ for a number of single particles, averaging it, and imposing that d| should correspond to the real diameter of a single particle. This enabled a smearing correction with submicron accuracy, without the need to measure the exposure time and local velocity independently. The correction for smearing is also illustrated in Figure 4. (24) Note: The real exposure time was shown to be slightly shorter than the (preset) interval between the images, because the images have to be stored on the PC hard disk directly after scanning. Studying the dependence of the degree of smearing Ds for the singlets as a function of their velocity, we found that it can be described by linear expression Ds ) vδt with the real exposure time δt ) 58 ( 3 ms (see Figure 5 for an illustration). Here v is the fluid velocity in the plane of observation. It should be noted that this quantity varies with the position above the bottom of the cell (z), as well as with the radial position (r). Measuring it for single particles at various r, z positions revealed that v showed a quantitative correspondence with r and z, as expected for affine motion: v ) zγ˘ ) z(ωr/H), with H the plate-plate distance in the shear cell.

as would hold in the case of circular cross-sections. Inspection of many (i.e., hundreds of) (d|, d⊥) combinations revealed no clear evidence for elongated aggregates (as might have been the case). Rejection of Off-Focus Aggregates. Another problem that had to be addressed was that entities which were clearly off the focal plane are still detected (albeit with a lower intensity). Incorporating these entities in the analysis would obscure the interpretation in a number of respects. First of all, not all object areas would correspond to cross sections anymore. Second, structures off the focal plane are moving with a different velocity and hence should require a different correction for “smearing out”. And finally, since large entities show a much stronger fluorescence than single particles, they are detected up to larger distances off the focal plane. Off-focus contributions could be largely eliminated by plotting all observed entities in an intensity-vs-area graph. In this graph, it was


Figure 6. Discrimination between aggregates that are in focus (b) and out of focus (O). The solid line (linear in a lin-lin plot) separating the two families was verified by inspection of the images. The dashed line shows the first intensity threshold, applied for noise rejection.

generally possible to distinguish two “families” of data, with the off-focal plane data belonging to the family of points which had a substantially smaller intensity per area (compared to the other family). This is also illustrated in Figure 6. III.7.2.2. Size Inaccuracies, and Histogram Interpretation. Inaccuracies in the Measured Cross Sections. There are two sources for inaccuracy in the aggregate size measuring procedure described above. The first one is related to the recognition of the object boundaries and the spread function of the CSLM. From Figure 7 it is clear that the measured size of a single particle will depend on the selected intensity threshold. This also applies to aggregates, but in a different way. If for example the threshold value is selected such that an isolated single particle is measured to have the proper size (known from the TEM characterizations), then the size of an aggregate after the same thresholding will be slightly larger than its real size. This is because in an aggregate, there are superpositions of the intensity profiles from other (nearby) particles. The error due to this effect amounts a fraction of the diameter of a single particle, and as such its importance becomes small for aggregates with D . 1 µm. The second source of inaccuracy is that the cross sections of aggregates on the CSLM image may belong to objects, whose centers of mass are actually located at different vertical positions and which are thus moving with different linear velocities. This causes an error in the correction for smearing, which becomes most pronounced for large aggregates moving close to the bottom of the cell: here, velocity differences between the plane of observation (γ*z) and the plane of the aggregate mass center (γ*(z+∆z)) become relatively large. For our experiments, conducted at z > 30 µm and for aggregates with D < 20 µm, this error was in most cases insignificant. Relation between Diameters and Cross Sections. The narrowness of the “sphere spread function” depicted in Figure 7, together with our selection of the intensity threshold, makes that in the case of aggregates, not the whole structure is seen but just an optical cross section. As a consequence hereof, there will be a distribution of cross-sectional areas even if all aggregates would be of identical size and spherically symmetric. The average of this distribution will underestimate the real size. To investigate the extent to which the observed histogram of cross sections deviates from the true size histogram, we have done a model calculation. Herein we

Langmuir, Vol. 20, No. 7, 2004 2621

have taken a Gaussian volume (i.e., φ325) distribution of homogeneous spheres, with an average diameter of 10 µm and a standard deviation of 2.5 µm. To simulate the effect of the optical sectioning of the CSLM, each point of this distribution was subjected to its corresponding “spread function”. This function (of general form f(d/D), with the argument (d/D) defined in the range 0 to 1) distributes the probability of observing an object with true diameter D, over apparent diameters d. Figure 8 shows the result of the calculation for a Gaussian distribution of homogeneous spheres, with a relative standard deviation of 30% (as typically found in the experiments). The figure illustrates that the differences between the “observed” and the “true” size distributions are fairly modest. As a consequence of the sectioning, the averaged size is underestimated by 10%, while the relative standard deviation is overestimated by 4%. In the analysis to follow, the cross-sectional area histograms will be treated as if they were “true” φ3 size distributions. IV. Results and Discussion IV.1. Slow and Fast Aggregation. A remarkable observation was that even at similar sample compositions, the time scale of the aggregate formation turned out to differ between experiments. Most cases could be classified as either “fast” (= 20 min to reach the steady state) or “slow” (hours) aggregation. The least complicated experiments to interpret are the fast ones. Here the typical aggregation time was found in reasonable agreement with the expected value for aggregation without kinetic barriers, as would be expected for aggregation due to nonadsorbing polymer. Note: In the case of slow aggregation there must be a mechanism operative, which opposes the formation of a bond between two particles once they are close together (since the transport mechanism bringing the particles together is dictated by the Peclet number and the particle concentration, which are equal for the fast and slow experiments (see Tolpekin et al.20 for a more detailed discussion of the transport mechanisms for bringing particles together)). This hindrance of bond formation between particles might be due to enhanced (i.e., in addition to hard sphere) repulsions between the silica particles, which can create a barrier in the particle interaction potential, for example, electrostatic repulsions. We note here that very little is known about the electrostatic interactions of silica particles in Br/Me. Besides that, the question remains what could modulate the electrostatic repulsion. The surface chemistry of the silica could be altered by the adsorption of a compound which becomes accessible after some time (like a decomposition product of the bromoform, or material leached out of the silica core). This would however require an investigation in its own right, which is beyond the scope of this paper. IV.2. An Aggregation Curve (for Fast Aggregation). Breakup. A typical curve showing the evolution of aggregate formation (and breakup) is shown in Figure 9. Note that the time t ) 0 corresponds to the start of the recordings, not of the mixing. The sharp increase in the (volume) average diameter corresponds to theoretical expectations. For the case of shear-controlled aggregation of compact aggregates, an exponential behavior should be expected (see ref 20). Fitting the data before the leveling off, resulted in a quantitative description (not shown here) for this time interval, and with a characteristic time scale (25) Flesch, J. C., Spicer, P. T., Pratsinis, S. E. AIChE J. 1999, 45, 1114-1124.

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Figure 7. Intensity profiles in radial and z directions, as observed for (immobile) single silica particles. This will also be referred to as the “sphere spread function”. Solid lines represent Gaussian fits (with σ the standard deviation).

Figure 8. Comparison between two theoretical size histograms (note that φ3 is a volume probability). The solid line corresponds to a Gaussian (φ3) distribution of homogeneous spheres, with Dv ) 10 µm and σ ) 2.5 µm. The dashed line is the corresponding distribution of cross-sectional areas as obtained when dividing each sphere into infinitesimally thin slices and assigning equal probabilities for encountering slices. The latter distribution, which mimics the CSLM observations, can be approximated as a Gaussian (φ3) distribution with Dv ) 9.0 µm and σ ) 2.6 µm.

Figure 10. φ3 histograms obtained from subsequent time spans of the curve shown in Figure 9: solid line and solid symbols, 1200-2100 s; dashed line and open symbols, 2100-2900 s. Vertical lines indicate the average diameters; their attached horizontal error bars indicate uncertainty ranges. The difference between the (fitted) histograms is statistically insignificant.

time bins for longer times (see the distance between subsequent data-points) resulted only in a partial reduction of the noise. The typical number of entities counted per unit time at the beginning of the experiment was 400700 per second. Despite the close mass-density matching, the observation time for the steady state was in many cases still limited by gravity settling. This can be understood from the expression for the steady sedimentation velocity

vsed = ∆FgD2/18η

Figure 9. Development of a steady state for an aggregating system in shear flow: silica volume fraction, 0.001; PEG concentration, 3.2 g/L; shear rate, 3.5/s. The average aggregate diameter (volume weighed) becomes constant over time. The solid line is drawn to guide the eye.

close to the expected value for flocculation without any kinetic barrier. This corroborates our previous assertion that the “fast aggregation” curves lend themselves the best for interpretation. If the system was only to aggregate, then the curve should show a monotonically increasing slope with time. The observed leveling off thus provides an indirect proof that also breakup is occurring. We note here that the on-line observation of individual breakup events is difficult. A discussion hereof can be found in Appendix 1. Steady State. After 1200 s, the average aggregate diameter Dv reaches a value which remains fairly constant for another 1800 s. The larger scatter in the data (compared to the first stage of the aggregation) is due to the smaller number of entities observed per unit time. Using larger

with D the diameter of the aggregate (here assumed to be compact). For a single particle having an excess density of 0.01 g/mL, calculation predicts that it takes ≈400 s to sediment over 1 µm, but for an aggregate with D ) 14 µm (as in Figure 9) this time is reduced to ≈2 s/µm. From this number, a maximum observation time can be calculated by taking into account the height of the fluid column above the plane of observation (≈300 µm), yielding ≈600 s. (As soon as the observation plane becomes depleted from the largest aggregate species, the representativity of the image is lost. An increased concentration of settled aggregates at the bottom did in our case not interfere with observations much “deeper” into the fluid.) In reality the time available for detecting a steady state has also been found shorter (sometimes necessitating rejection of the experiment) or longer than expected (presumably due to uncontrolled small variations in the solvent density). The latter seems to be the case for the experiment shown in Figure 9. Since this was one of the longest running experiments without clearly noticeable sedimentation effects, we have used this experiment to inspect more critically whether a steady state was reached. This can be done by looking at the aggregate size distributions. Figure 10 shows two histograms, obtained by averaging over t ) 1200-2100 s and 2100-2900 s. Within the accuracy of the measurements, the size distributions are


Langmuir, Vol. 20, No. 7, 2004 2623

Figure 11. (left plot) Evolution of (volume-weighted) average aggregate size at a shear rate of 3.5/s, and PEG concentrations ranging from 2.7 to 3.9 g/L. Lines (solid lines corresponding to solid symbols) are drawn to guide the eye. (right plot) Average size in the steady state, as a function of PEG concentration. Solid symbols correspond to the experiments at 3.5/s and open symbols to higher shear rates. Table 1. Volume-Weighted Average Aggregate Size (µm) as a Function of cP and Shear Rate [PEG] ) 2.7 g/L [PEG] ) 3.0 g/L [PEG] ) 3.2 g/L [PEG] ) 3.9 g/L

γ ) 3.5/s

γ ) 5.6/s

10.4 ( 0.14 11.67 ( 0.34 14.57 ( 0.19 19.19 ( 0.54

7.26 ( 0.18 8.64 ( 0.11 10.10 ( 0.17

γ ) 6.3/s

10.48 ( 0.36

indistinguishable. While this may not be a rigorous proof for the occurrence of a steady state, it is the strongest corroboration thereof as possible with our experiments. IV.3. Dependence on Polymer Concentration. The PEG concentration was varied between 2 and 12 g/L, but above 4 g/L the aggregates grew too large to allow for reliable and representative measurements of the aggregate size distribution. The results obtained at lower concentrations are shown in Figure 11. IV.4. Dependence on Shear Rate. The dependence of the average aggregate size on shear rate was investigated by quantitative image analysis at different radial positions in the shear cell (3 and 8 or 9 mm), after completing the observations at 5 mm (shown in Figure 11). The observations at 3 mm invariably showed large aggregated networks, sedimented to the bottom of the cell: a dramatic difference, considering that the shear rate was only reduced by 40%. At 8 (9) mm the shear rate is 60 (80)% larger compared to 5 mm. The aggregate size (distributions) measured at these locations can be regarded as corresponding to a steady state, considering that a higher shear rate can only accelerate the aggregation process, while in addition the aggregates should not have to grow as large as at 5 mm: at higher shear rate the steady-state aggregate size is expected to be smaller. The shear rate dependence of the mean aggregate size is summarized in Table 1 (and plotted in Figure 11). Together with the absence of breakup at γ e 2/s (at r ) 3 mm) and the possibility to destroy all aggregates by simple hand shaking (see section III.5), it illustrates that the balance between aggregation and breakup is very subtle; i.e., the fluid used in this study is very sensitive to shear. Analysis of the dependence of the (volume averaged) aggregate size as a function of the PEG concentration and the shear rate shows that (within the accessible cPEG and γ ranges) the behavior can be mathematically represented by Dv ) 5.8(cPEG/γ˘ )5/3, with Dv in µm, cPEG in g/L, and γ˘ in s-1, as is also shown by Figure 12. IV.5. A Closer Look at Aggregate Size Distributions. IV.5.1. Transient Stages. Typical aggregate size distributions as measured in different stages of the

Figure 12. Aggregate sizes in the steady state, as a function of polymer concentration and shear rate, collapsed onto a master plot (log-log). Different symbols correspond to different shear rates. The slope of the line amounts to approximately 5/3.

Figure 13. Transient size distributions (right plot) for the experiment with 3.9 g/L PEG and shear rate 3.5/s. The left plot shows the times at which the distributions were sampled: 0 s, 500 s, and 800-1400 s. Symbols in the right and left plots correspond to each other. Solid lines are drawn to guide the eye. Comparison of the steady state distributions at different heights confirmed the absence of a bias due to sedimentation of the largest aggregates. This implies that also the transient distributions must be unbiased, since there the aggregates are smaller while also the time is shorter.

aggregation are shown in Figure 13. The distributions are significantly broader than what could be ascribed to the slicewise sampling of the CSLM (as discussed in section III.7.2.2). Therefore, we think that the broadness of the histograms signifies the probabilistic nature of aggregate formation and breakup: even if only one aggregate size would be mechanically stable, there will always be smaller and larger aggregates present as well. This is because two undersized aggregates need time to join, and an oversized aggregate may need to be exposed to the flow for sufficiently long (in its “weakest orientation”). Apart from these considerations, also differences in the aggregate

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Figure 14. Normalized histograms in the steady state: left, γ˘ ) 3.5 s-1; right, 6 s-1. Circles and the solid line correspond to 2.7 g/L, squares and dashed line to 3.0 g/L, diamonds and dash-dotted line to 3.2 g/L, and triangles and dash-double dotted line to 3.9 g/L experiments.

Figure 15. Relative standard deviations, obtained from a Gaussian fit to the φ3 histograms, plotted against PEG concentration: solid symbols, γ˘ ) 3.5 s-1; open symbols, 6 s-1.

structures and in the strength of the “critical link” (e.g., the number of particle-particle bonds involved in the breakup2,3) may play a role. These factors should further broaden the distribution. While it is not shown in this paper, we have also compared the shapes of the size distributions (after normalizing them) in the transient part of the curves. It turned out that, except for the early aggregation stages (D , Dfinal) the shapes are rather similar. This selfpreserving property justifies the approach described in section II to analyze the population balance model (see also section IV.7). IV.5.2. Steady State Size Distributions. The aggregate size histograms in the steady state can generally be obtained with higher accuracy, owing to the longer period over which they can be measured. After multiplying the horizontal axis with 1/Dv and normalizing the distribution for each of the histograms, it turns out that all distributions have a similar shape and width. This is illustrated in Figures 14 and 15 for the two most occurring shear rates. It indicates that the trends with PEG concentration and shear rate, as observed for the average diameter, are not too sensitive to the particular choice that was made for volume weighted averages. Again, the typical standard deviation of 25% is too large to be attributed to the optical sectioning of the CSLM. IV.6. Structure of the Aggregates. Stopped flow experiments turned out to be well-suited for investigating aggregate structures, without being hampered by the smearing effects of the flow, and without “memory loss” for the preceding flow history. First of all we observed that (careful) slowing down the flow to zero did not result in aggregate breakup. Second, the time lapsed after stopping the flow and making recordings is much too short to change the aggregate size distribution as created by the flow history (i.e., the prolonged constant shear rate).

We note here that in a paper in 2002 26 it was remarked that confocal microscopy is an unsuitable technique for examining the detailed structure (of a freely moving particle aggregate), due to the mobility of the structure (elements) and the long exposure time. In our case, the exposure time could be kept very short (58 ms), which did allow us to closely examine some aggregates. We performed z-scans (imaging at different vertical positions, with a step size of 1 µm) for two selected aggregates. Representative (except perhaps for some anisotropy in panels a-c) cross-section images are shown in Figure 16. Obviously, the image quality is much better now than in recordings such as Figure 3. We recall here that the number of aggregates that can be measured per unit time in a stopped-flow situation is too small to allow a timeresolved measurement of the aggregation process. We characterized the structure of selected aggregates by measuring the fractal dimension. First, individual particle positions (xi, yi) and intensities I0,i were determined for each z-slice with help of the procedure described in section III.7. We considered two possible ways to calculate the fractal dimension for given set of particle positions. The first one involves reconstruction of 3D particle coordinates (xi, yi, zi) from the data for each z-slice, counting the number of particles, N(R), inside the spheres of different radii, R, and comparing the result with a fractal law

(Ra)

N(R) ) N0

Df

(IV.6.1)

with Df the fractal dimension, N0 a numerical constant, and a the radius of a primary particle. The second way is based on counting the particles inside the circles of different radii, r, for different z-slices (of fixed width ∆z) and comparing it to the calculated dependence of N(r,z). We selected the second possibility. The derivation for N(r,z) in cylindrical coordinates (z, r, φ) is given in Appendix 2. The coordinate center is put to the center of mass of the aggregate. Introducing dimensionless variables, F ) r/a and ξ ) z/a for thin slices (∆z , R0, where R0 is a radius of the aggregate), one obtains

N(F,) ≈ N0

∆z Df [(F2 + ξ2)(Df-1)/2 - ξDf-1] a Df - 1 (IV.6.2.)

Here ∆z is the slice thickness. This quantity was regulated with the intensity threshold (see also Figure 7) (26) Bushell, G. C.; Yan, Y. D.; Woodfield, D.; Raper, J.; Amal, R. Adv. Colloid Interface Sci. 2002, 951-50.


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Figure 16. Typical cross sections of aggregates in the steady state, after stopping the shear flow. Images a, b, and c correspond to the cross sections made at -5, 0 and +5 µm from the aggregate center-of-mass plane. Image d corresponds to central cross section of a different aggregate. Scale bar: 10 µm.

[ ( )]

I ) Io exp -

1 ∆z 2 σz

IV.7. Modeling. In section II an expression for the average aggregate size 〈R/a〉 as a function of time was derived

2

Equation IV.6.2 was used to find Df by fitting N(F,). For a central slice, ξ ) 0, and expression IV.6.2 simplifies to

N(F) ) N0

∆z Df FDf-1 a Df - 1

(IV.6.3.)

The results for the measured N(r,z) functions are presented in Figure 17. We first fitted the data for ξ ) 0 as these contained the highest number of particles (thus giving the statistically most relevant fit). Only the data in the middle F range (8 < F < 20) were used in fitting because for lower F values N(r) is low (and thus scattered) and for the highest values a “saturation” in N(r) occurs, which is related to approaching of the boundary of the aggregate. The results of the fit are

Df ) 2.1 ( 0.1, N0 ) 0.70 ( 0.2 (Figure 17a) and

Df ) 2.0 ( 0.1, N0 ) 0.55 ( 0.1 (Figure 17b) Using these data we plotted the functions N(r) for other two slices with z ) (5 µm. The agreement with the data turns out to be satisfactory (albeit that the r-range does not cover more than 1 decade). The value of 2.0 corresponds fairly well with earlier studies. As discussed in an earlier paper3 one mostly finds for shear controlled aggregation Df ) 2.2 ( 0.2. Moreover, from our flow curve analysis we found in the past Df ) 2.4 ( 0.1 (ref 4) and Df ) 2.3 ( 0.1 (ref 2) for the shear controlled case (due to compaction).

d 1 〈R/a〉 ) (β1Bo〈R/a〉4-Df - β2Ko〈R/a〉Q+1) dt Df

(II.7)

In this derivation, self-preservation of the size distribution was assumed. In section IV.5.2 it was indicated that the volume distributions are Gaussian and indeed almost self-preserving

φ3(D) ) (2π)-1/2/σv exp(-1/2(D - Dv)2/σv2) (IV.7.1) with 0.2 < σv/Dv < 0.3 independent of time. In the sequel we take σv/Dv ) 0.25, which corresponds to σr ) σo/Ro ) 0.37 for the number distribution as used in section II. With this knowledge, values for β1 and β2 can be calculated as well as β3 ) 〈D/a〉v/〈R/a〉 ) 2〈(R/a)4〉/(〈(R/a)3〉〈R/a〉)

β1 ) 1.06

β2 ) 1.16

β3 ) 2.67

To compare the model calculations with the experiments, the value for Df was set to 2.0, in accordance with the results from section IV.6, and the time constant for aggregation, to ) Df/β1Bo, was calculated: to ) 460 s. The only unknowns left in eq II.7 were Ko and Q, which have been used as a fitting parameter, when the results of the numerical integration of eq II.7 were compared with the experimentally obtained curves. We have allowed Ko and Q to be a function of the polymer concentration cP. Ko can be expected to depend on cP, since for higher cP, a larger force is needed to break a particle bond. Allowing Ko to vary, and requiring the model curves to fit the data, it was found that also Q has to be depend on cP. We consider this finding as a reflection of the physical behavior of our specific system.

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Figure 17. Data fitting, assuming a fractal law, of radial distribution functions obtained from the stopped-flow images. Plots a and b correspond to parts a-c of Figure 16 and Figure 16d, respectively. Circles (a, b), squares (b), and triangles (b) correspond to the z-slices of 0, -5, and +5 µm, respectively. The slice thickness ∆z ) 1.37a, with a the radius of a single particle.

the dependences with power laws, Ko scales as cp-2 and Q as cp1.3. Assuming that the particle-particle interaction energy, U/kT, scales as cp (as for a pair potential dominated by polymer depletion, see section III.4, but see also the note in section IV.1), this would imply that Ko ∝ (U/kT)-2 and Q ∝ (U/kT)1.31. V. Conclusions

Figure 18. The average aggregate size as a function of time for four different polymer concentrations. The points indicate the experimental results while the curves result from a fit of the model to the experimental data; Ko and Q were fitted.

Figure 19. The fitted breakup rate Ko ) 1/tesc and exponent Q as a function of the PEG concentration. The points represent the fitting results, while the lines are power law fits: Ko ) 3.1 × 10-4 (cp/g L-1)-2 and Q ) 0.43 (cp/g L-1)1.31.

The outcome of this procedure is given in Figure 18 where the calculated curves are compared with the experimental results. The fits are in quantitative agreement with the data. In Figure 19 the corresponding breakup rate Ko and the exponent Q have been plotted as a function of the PEG concentration cp. Characterizing

For the first time, aggregation curves of colloidal particle aggregates in shear flow have been measured in real time and space with a CSLM. This was achieved using a stateof-the-art confocal microscope and after developing a solvent in which the colloidal particles are matched with respect to both refractive index and mass density. The latter allowed study of the aggregation process from the beginning up until the steady state, in which aggregation and breakup events maintain a dynamic balance. The method presented in this paper is complementary to scattering methods, which allow better statistics via their larger sampling volume but provide indirect information. Our method does not rely on any model to interpret the data in terms of structures but measures the structures directly. Moreover, the direct observation allows recognition of various possible complications (potentially present also in other types of experiments), which can be difficult to detect otherwise. Among these are deviations from system homogeneity or from the average flow field. Also the distinction between the behavior of the suspension in bulk or near a surface can be made. For the newly developed particle system, very weak aggregates were be obtained by adding a nonadsorbing polymer as a flocculation agent. The aggregates were so weak that a gentle shaking with the hand was enough to redisperse all particles as individual entities. A quantitative characterization of flow-induced aggregation and breakup was obtained by performing well-defined experiments in shear flow. Parameter variation showed that the average aggregate size depends sensitively on a competition between cohesive forces caused by the polymer and rupture forces caused by the flow. The precise mode(s) for aggregate breakup could not be elucidated due to an insufficient number of detectable breakup events. However, information about breakup is contained in the structures and size distribution of the aggregates in the steady state, both of which have been obtained (for the first time) in this study. The aggregate


size distributions (in the steady state) were shown to be broad, with dispersities up to 30%. This demonstrates the probabilistic nature of aggregation and breakup events, as opposed to the simplistic (yet successful) model in which one kind of aggregate is supposed to occur, with the size determined by a force balance. The structure of the aggregates turned out to be fractal, with a fractal dimension of 2.0-2.1. This could be concluded from observations after stopping the flow. The possibility to study structures of objects (defined by dynamic processes), in the quiescent state, was owing to the much shorter time scale for image capture compared to the time needed for an aggregate to displace itself, or change its structure in the absence of flow. Together with kinetic coefficients for (Brownian and shear-induced) aggregation, the fractal dimension of the aggregates was used in a population balance model. With this population balance model, it was possible to describe the set of aggregation curves quantitatively. Acknowledgment. We thank Jasper Overman for characterization of the system and preliminary experiments during his graduation study at the Physics of Complex Fluids group. This work has been supported by the Foundation for Fundamental Research on Matter (FOM), which is financially supported by The Netherlands Organization for Scientific Research (NWO). Appendix 1 This appendix serves to illustrate that on-line observation of individual breakup events is difficult in experiments as described in this paper. Assuming that the particles move along with the laminar flow pattern as defined by the shear flow, the rate of observed particle encounters is estimated as 2

2 3

9φ HLDγ˘ /8π a ) 0.03/s with φ the volume fraction, γ the shear rate, a the particle radius, LD the area seen by the microscope, and H the z-range of the focus (as defined by the optics and a rejection criterion in the image processing). This very low rate becomes even smaller when encounters between aggregates are considered: then the given expression has to be multiplied with 3

Langmuir, Vol. 20, No. 7, 2004 2627

for a′ ) 5 µm and τ = 200 s (see Figure 11). These approximate calculations show that both aggregation and breakup events are so rarely observed that there is no point looking for them (in order to make an analysis of details such as when and how aggregates breakup). This is due to the cooperative effect of: small sampled space and small volume fraction. Aggregation makes this even worse since it gives a further reduction in the number of entities. Appendix 2 In this appendix the expression for N(r,z), the number of particles in a disk of radius r at height z above the centerr-of-mass plane of a fractal aggregate, is derived in cylindrical coordinates (z, r, φ). The number of particles N inside the spherical aggregate with radius R is given by

N(R) ) N0(R/a)Df Assuming an isotropic particle distribution, we can write for the particle density

n(R) )

for a′ ) 5 µm (see Figure 11) with H′ the effective z-range for detecting aggregates, a′ the radius of the aggregate, and φagg (=0.5) the particle volume fraction within the aggregate. To estimate the rate of observed breakup events, one should first consider the average number of aggregates per microscope image. This number is given by

M)

3φH′LD ) 0.18 4πφagga′3

for a′ ) 5 µm. Supposing that aggregate breakup is an autonomous event, characterized by a exp(-t/τ) dependence, the typical rate of observed breakups becomes

M/τ = 0.001/s

Df-3

()

The number of particles inside a disk (in cylindrical coordinates)

0 < r′ < r z - ∆z < z′ < z + ∆z 0 < φ < 2π is then found by integrating over the disk volume

N(z,r) )

z+∆z r dz′ ∫0 n(z′,r′)r′ dr′ ∫02π dφ ∫z-∆z

It is convenient to choose dimensionless coordinates

F ) r/a,

ξ ) z/a

The expression for N(z,r) then becomes

3

2 a H′ a 1 = = 0.005 H a′3 φagg H a′2

dN N0Df R ) dV 4πa3 a

N(ξ,F) )

N0Df

ξ+∆ξ [F2 + ξ′2](D -1)/2 dξ′ ∫ξ-∆ξ

{

2(Df - 1)

f

ξ+∆ξ [ξ′2](D -1)/2 dξ′} ∫ξ-∆ξ f

The first integral in the previous expression can in principle be rewritten with a hypergeometric function. However, we prefer to use a more simple approximate expression for thin disks

∆z , R0 where R0 is a radius of the aggregate. In this case the integrals are replaced with the average value multiplied by the width of the integration range. This gives the final expression for N(ξ,F)

N(ξ,F) ) N0 LA035758L

Df ∆ξ{[F2 + ξ2](Df-1)/2 - ξDf - 1} Df - 1

Aggregation and Breakup of Colloidal Particle ... - ACS Publications

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